Can a function be increasing *at a point*?
A function $f$ is increasing at $x$ if $f(t)\gt f(x)$ for every $t\gt x$ close enough to $x$ and $f(t)\lt f(x)$ for every $t\lt x$ close enough to $x$. More rigorously, one asks that there exists $\varepsilon\gt0$ such that, for every $(t,s)$ such that $x-\varepsilon\lt t\lt x\lt s\lt x+\varepsilon$, $f(t)\lt f(x)\lt f(s)$.
No notion of differentiability is needed. Consider for example the function $f$ defined by $f(t)=2t$ for every rational $t$ and $f(t)=t$ for every irrational $t$. Then $f$ is increasing at $x=0$ and only at $x=0$ while $f$ is nowhere differentiable.